What is the general solution for csc(x)= 1/(cos(x)) ?

The general solution for csc(x)= 1/(cos(x)) is x= pi/4+pin

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csc(x)= 1/(cos(x))

csc (x) = \frac{1}{cos ( x )}

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

csc (x) = \frac{1}{cos ( x )}

Subtract

\frac{1}{cos ( x )}

from both sides

csc (x) - \frac{1}{cos ( x )} = 0

Simplify

csc (x) - \frac{1}{cos ( x )} : \frac{csc ( x ) cos ( x ) - 1}{cos ( x )}

csc (x) - \frac{1}{cos ( x )}

Convert element to fraction:

csc (x) = \frac{csc ( x ) cos ( x )}{cos ( x )}

= \frac{csc ( x ) cos ( x )}{cos ( x )} - \frac{1}{cos ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{csc ( x ) cos ( x ) - 1}{cos ( x )}

\frac{csc ( x ) cos ( x ) - 1}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

csc (x) cos (x) - 1 = 0

Rewrite using trig identities

csc (x) cos (x) - 1

csc (x) cos (x) = cot (x)

csc (x) cos (x)

Express with sin, cos

csc (x) cos (x)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( x )} cos (x)

Simplify

\frac{1}{sin ( x )} cos (x) : \frac{cos ( x )}{sin ( x )}

\frac{1}{sin ( x )} cos (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 cos ( x )}{sin ( x )}

Multiply:

1 \cdot cos (x) = cos (x)

= \frac{cos ( x )}{sin ( x )}

= \frac{cos ( x )}{sin ( x )}

= \frac{cos ( x )}{sin ( x )}

Use the basic trigonometric identity:

\frac{cos ( x )}{sin ( x )} = cot (x)

= cot (x)

= - 1 + cot (x)

- 1 + cot (x) = 0

Move

1

to the right side

- 1 + cot (x) = 0

Add

1

to both sides

- 1 + cot (x) + 1 = 0 + 1

Simplify

cot (x) = 1

cot (x) = 1

General solutions for

cot (x) = 1

cot (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

csc (θ) = (1.201)

3 cos (θ) + 2 = 5 sec (θ), 0 \leq θ < 2 π

2 cos (\frac{π}{4} - 3 x) = 2

3 sec^{2} (x) = sec (x)

tan^{2} (x) - 1 = sec^{2} (x)

What is the general solution for csc(x)= 1/(cos(x)) ?
The general solution for csc(x)= 1/(cos(x)) is x= pi/4+pin